MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tanatan Structured version   Unicode version

Theorem tanatan 22336
Description: The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
tanatan  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )

Proof of Theorem tanatan
StepHypRef Expression
1 atancl 22298 . . 3  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
2 2efiatan 22335 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
32oveq1d 6127 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 ) )
4 2mulicn 10569 . . . . . . . 8  |-  ( 2  x.  _i )  e.  CC
54a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6 atandm 22293 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
76simp1bi 1003 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  A  e.  CC )
8 ax-icn 9362 . . . . . . . 8  |-  _i  e.  CC
9 addcl 9385 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
107, 8, 9sylancl 662 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
11 subneg 9679 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
127, 8, 11sylancl 662 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
136simp2bi 1004 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
148negcli 9697 . . . . . . . . . 10  |-  -u _i  e.  CC
15 subeq0 9656 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
1615necon3bid 2637 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
177, 14, 16sylancl 662 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
1813, 17mpbird 232 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
1912, 18eqnetrrd 2656 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
205, 10, 19divcld 10128 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  e.  CC )
21 ax-1cn 9361 . . . . . 6  |-  1  e.  CC
22 npcan 9640 . . . . . 6  |-  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 )  +  1 )  =  ( ( 2  x.  _i )  / 
( A  +  _i ) ) )
2320, 21, 22sylancl 662 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i )
) )
243, 23eqtrd 2475 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i ) ) )
25 2muline0 10570 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
2625a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =/=  0 )
275, 10, 26, 19divne0d 10144 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  =/=  0
)
2824, 27eqnetrd 2654 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )
29 tanval3 13439 . . 3  |-  ( ( (arctan `  A )  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )  ->  ( tan `  (arctan `  A ) )  =  ( ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
301, 28, 29syl2anc 661 . 2  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
312oveq1d 6127 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 ) )
3221a1i 11 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  1  e.  CC )
3320, 32, 32subsub4d 9771 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
1  +  1 ) ) )
34 df-2 10401 . . . . . . . 8  |-  2  =  ( 1  +  1 )
3534oveq2i 6123 . . . . . . 7  |-  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
1  +  1 ) )
3633, 35syl6eqr 2493 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  2 ) )
3731, 36eqtrd 2475 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
38 2cn 10413 . . . . . . . 8  |-  2  e.  CC
39 mulcl 9387 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( A  +  _i )  e.  CC )  ->  ( 2  x.  ( A  +  _i )
)  e.  CC )
4038, 10, 39sylancr 663 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( A  +  _i ) )  e.  CC )
415, 40, 10, 19divsubdird 10167 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) ) ) )
42 mulneg12 9804 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  =  ( 2  x.  -u A
) )
4338, 7, 42sylancr 663 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  =  ( 2  x.  -u A ) )
44 negsub 9678 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
458, 7, 44sylancr 663 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
4645oveq1d 6127 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  ( ( _i 
-  A )  -  _i ) )
477negcld 9727 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  -u A  e.  CC )
48 pncan2 9638 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( ( _i  +  -u A )  -  _i )  =  -u A
)
498, 47, 48sylancr 663 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  -u A )
508a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5150, 7, 50subsub4d 9771 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  -  A )  -  _i )  =  ( _i  -  ( A  +  _i )
) )
5246, 49, 513eqtr3rd 2484 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( _i 
-  ( A  +  _i ) )  =  -u A )
5352oveq2d 6128 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( 2  x.  -u A
) )
5438a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5554, 50, 10subdid 9821 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( ( 2  x.  _i )  -  (
2  x.  ( A  +  _i ) ) ) )
5643, 53, 553eqtr2rd 2482 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  =  ( -u 2  x.  A ) )
5756oveq1d 6127 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( -u 2  x.  A )  /  ( A  +  _i )
) )
5854, 10, 19divcan4d 10134 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) )  =  2 )
5958oveq2d 6128 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( 2  x.  ( A  +  _i )
)  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6041, 57, 593eqtr3d 2483 . . . . 5  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  ( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6137, 60eqtr4d 2478 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( (
-u 2  x.  A
)  /  ( A  +  _i ) ) )
6224oveq2d 6128 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
638, 38, 8mul12i 9585 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  _i ) )  =  ( 2  x.  (
_i  x.  _i )
)
64 ixi 9986 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
6564oveq2i 6123 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  _i ) )  =  ( 2  x.  -u 1
)
6621negcli 9697 . . . . . . . . 9  |-  -u 1  e.  CC
6738mulm1i 9810 . . . . . . . . 9  |-  ( -u
1  x.  2 )  =  -u 2
6866, 38, 67mulcomli 9414 . . . . . . . 8  |-  ( 2  x.  -u 1 )  = 
-u 2
6963, 65, 683eqtri 2467 . . . . . . 7  |-  ( _i  x.  ( 2  x.  _i ) )  = 
-u 2
7069oveq1i 6122 . . . . . 6  |-  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  (
-u 2  /  ( A  +  _i )
)
7150, 5, 10, 19divassd 10163 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i ) ) ) )
7270, 71syl5eqr 2489 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
7362, 72eqtr4d 2478 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( -u 2  /  ( A  +  _i ) ) )
7461, 73oveq12d 6130 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) ) )
7538negcli 9697 . . . . . 6  |-  -u 2  e.  CC
76 mulcl 9387 . . . . . 6  |-  ( (
-u 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  e.  CC )
7775, 7, 76sylancr 663 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  e.  CC )
7875a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  e.  CC )
79 2ne0 10435 . . . . . . 7  |-  2  =/=  0
8038, 79negne0i 9704 . . . . . 6  |-  -u 2  =/=  0
8180a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  =/=  0 )
8277, 78, 10, 81, 19divcan7d 10156 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  ( (
-u 2  x.  A
)  /  -u 2
) )
837, 78, 81divcan3d 10133 . . . 4  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  -u 2
)  =  A )
8482, 83eqtrd 2475 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  A )
8574, 84eqtrd 2475 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  A )
8630, 85eqtrd 2475 1  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   dom cdm 4861   ` cfv 5439  (class class class)co 6112   CCcc 9301   0cc0 9303   1c1 9304   _ici 9305    + caddc 9306    x. cmul 9308    - cmin 9616   -ucneg 9617    / cdiv 10014   2c2 10392   expce 13368   tanctan 13372  arctancatan 22281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-tan 13378  df-pi 13379  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-haus 18941  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-tms 19919  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030  df-atan 22284
This theorem is referenced by:  atantanb  22341  atanord  22344
  Copyright terms: Public domain W3C validator